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MathGroup Archive 2006

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problems with sum functions/ factoring the factorial

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65698] problems with sum functions/ factoring the factorial
  • From: Roger Bagula <rlbagulatftn at yahoo.com>
  • Date: Sun, 16 Apr 2006 01:44:55 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I have no problem constructing the Primorial or Compositorial functions 
that factor the factorial as:
cf[n]*p[n]=n!
But trying to get the sine and cosine constructed functions to plot
seems to be a problem here:
Clear[f, g, cf, p, CeS, CeC, PeS, PeC]
f[n_] := If[PrimeQ[n] == True, 1, n]
cf[0] = 1;
cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1]
g[n_] := If[PrimeQ[n] == True, n, 1]
p[0] = 1;
p[n_Integer?Positive] := p[n] = g[n]*p[n - 1]
Ce = 1 + Sum[1/cf[n], {n, 1, 1000}];
N[%, 100]
Pe = 1 + Sum[1/p[n], {n, 1, 1000}];
N[%, 100]
CeS[x_] := 1 + NSum[(-1)^n*p[2*n + 1]*x^(2*n + 1)/(2*n + 1)!, {n, 1, 100}];
CeC[x_] := 1 + NSum[(-1)^n*p[2*n]*x^(2*n)/(2*n)!, {n, 1, 100}];
ParametricPlot[{CeC[x], CeS[x]}, {x, 0, 2*Pi}]
PeS[x_] := 1 + NSum[(-1)^n*cf[2*n + 1]*x^(2*n + 1)/(2*n + 1)!, {n, 1, 100}];
PeC[x_] := 1 + NSum[(-1)^n*cf[2*n]*x^(2*n)/(2*n)!, {n, 1, 100}];
ParametricPlot[{PeC[x], PeS[x]}, {x, 0, 2*Pi}]

Alernative functions:
CeS1[x_] := 1 + NSum[(-1)^n*x^(2*n + 1)/cf[2*n + 1], {n, 1, 100}];
CeC1[x_] := 1 + NSum[(-1)^n*x^(2*n)/cf[2*n], {n, 1, 100}];
ParametricPlot[{CeC[x], CeS[x]}, {x, 0, 2*Pi}]
PeS1[x_] := 1 + NSum[(-1)^n*x^(2*n + 1)/p[2*n + 1], {n, 1, 100}];
PeC1[x_] := 1 + NSum[(-1)^n*x^(2*n)/p[2*n], {n, 1, 100}];
ParametricPlot[{PeC1[x], PeS1[x]}, {x, 0, 2*Pi}]

In addition this function seem to come up with the wrong sign:
Pe[x_] := 1 + NSum[cf[n]*x^n/n!, {n, 1, 100}];
Plot[Pe[x], {x, 0, 5}]

Or alternatively:
Pe1[x_] := 1 + NSum[x^n/p[n], {n, 1, 100}];
Plot[Pe1[x], {x, 0, 5}]

Ce*Pe~ 5*E (low)
Roger


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